目的：使用遗传算法寻找方程的最大值

这里先放最后的结果：，可以看到在经过多次迭代后，初始种群在向函数的最大值点逼近。
遗传算法是用于解决NP—hard问题的，目的是为了有效的求解出相对最优的解，用计算机上的语言来说就是以损失一定精度换取求解时间。在网上有很多描述遗传算法的，道理很简单，就是模拟八个字“物竞天择，优胜劣汰”（这里不得不佩服John Henry Holland这位大师的脑洞），学完遗传算法，蚁群算法，粒子群算法，模拟退火算法我还是想感叹一句，生活就是最美的数学。话不多说，接下来说一下我的思路。

步骤

1-确定解码方式（我采用的是二进制编码，因为更形象，也简单），取精度为0.0001，函数自变量x的范围为[0,9],那么需要的位数为17位，所以解码代码如下：

#解码
def decode(genCode):
    y = 0 + int(genCode,2)/(2**17-1)*9    
    return y

2-初始化种群（遗传算法的效果受初始化种群的影响较大，选择不好很容易陷入局部最优）

#初始化种群数(假设初始种群数目为10个)，10个17位位str
population = [] #总群
for i in range(20):
    strCode = ''
    for j in range(17):
        strCode += str(np.random.randint(0,2))
    population.append(strCode)

3-适应函数（这里就是目标函数），求解每个个体的适应值

def fitnessfunction(population,initFunction):
    value = []
    for i in range(len(population)):
        value.append(initFunction(decode(population[i])))
        if value[i] < 0:
            value[i] = 0
    return value

4-轮盘赌选择（模拟优胜劣汰），根据个体适应值大小，采用轮盘法选择优胜个体

def selectChild(population,fitnessResult):
    childPopulation = []
    Sumfitness = []
    indiviSumP = []
    for i in range(len(fitnessResult)):
        if i == 0:
            Sumfitness.append(fitnessResult[i])
        else:
            Sumfitness.append(Sumfitness[i-1] + fitnessResult[i])   
    for j in range(len(Sumfitness)):
        indiviSumP.append(Sumfitness[j]/sum(fitnessResult))
    for childNum in range(len(population)):
        x = np.random.uniform(0,1)
        if x <= indiviSumP[0]:
            childPopulation.append(population[0])
        else:
            for m in range(1,len(population)):
                if indiviSumP[m-1] <= x and x <= indiviSumP[m]:
                    childPopulation.append(population[m])
                    break     
    return childPopulation

5-交叉选择（对选择的优胜个体进行交叉信息，模拟繁殖下一带，且用一个概率控制满足自然界中颜色体交换是普遍的（所以这里的交叉概率取固定的0.7），变异是罕见的（这里取较小0.002））

#对选择后的子代进行一定概率的染色提交换
def crossParent(new_population,pc):
    crossChildPopulation = []
    #如果为单身则先选择出单shen
    if len(new_population)%2 != 0:
        randmSingleNum = np.random.randint(0,len(new_population))
        crossChildPopulation.append(new_population[randmSingleNum])
        del(new_population[randmSingleNum])

    half = int(len(new_population)/2)
    father = new_population[:half]
    mother = new_population[half:]
    np.random.shuffle(father)
    np.random.shuffle(mother)

    for i in range(half):
        crossP = np.random.uniform(0,1)
        if pc >= crossP:
            breakPoint = np.random.randint(0,len(new_population[0]))
            son = father[i][:breakPoint] + mother[i][breakPoint:]
            daughter = mother[i][:breakPoint] + father[i][breakPoint:]
        else:
            son = father[i]
            daughter = mother[i]
        crossChildPopulation.append(son)
        crossChildPopulation.append(daughter)
    return crossChildPopulation

6-变异获得子代

#变异
def variation(crossChildPopulation,pc):
    variationPopulation = []
    for i in range(len(crossChildPopulation)):
        variationP = np.random.uniform(0,1)
        variationPos = np.random.randint(0,len(crossChildPopulation[0]))
        if variationP <= pc:
            if variationPos == 0:
                if crossChildPopulation[i][0] == '0':
                    crossChildPopulation[i] = '1'+crossChildPopulation[i][1:]
                else:
                    crossChildPopulation[i] = '0'+crossChildPopulation[i][1:]
            else:
                if crossChildPopulation[i][variationPos] == '0':
                    crossChildPopulation[i] = crossChildPopulation[i][:variationPos-1] + '1' + crossChildPopulation[i][variationPos:]
                else:
                     crossChildPopulation[i] = crossChildPopulation[i][:variationPos-1] + '0' + crossChildPopulation[i][variationPos:]
        variationPopulation.append(crossChildPopulation[i])
    return variationPopulation

然后进行多步训练即可：

for i in range(1000):
    fitnessValue = fitnessfunction(population,initFunction)
    selectChildes = selectChild(population,fitnessValue)
    crossChild = crossParent(selectChildes,0.7)
    population = variation(crossChild,0.02)

但是我们想要模拟动态过程，所以我们需要引入

from matplotlib.animation import FuncAnimation
from matplotlib import animation

用于绘制我们的动图，如前面所示。这里大家可以取查官方文档看这个的具体用法，或者给我留言，我就直接贴更新图片获得动图的代码：

def updata(i):
    global population
    x1 = [0 for j in range(len(population))]
    y1 = [0 for j in range(len(population))]
    fig.set_size_inches(15,10)#控制再保存图片的时候窗口不要缩小
    picpath = 'C:\\Users\\Amie\\Desktop\\png\\{0}.png'.format(i)
    plt.savefig(picpath,pi = 600,edgecolor = 'r')

    fitnessValue = fitnessfunction(population,initFunction)
    selectChildes = selectChild(population,fitnessValue)
    crossChild = crossParent(selectChildes,0.7)
    population = variation(crossChild,0.02)
    for i in range(len(population)):
        x1[i] = decode(population[i])
        y1[i] = initFunction(decode(population[i]))
    data = [[x,y] for x,y in zip(x1,y1)]
    sca1.set_offsets(data)
    print(list(zip(x1,y1)))
    return sca1

现在我们只需要组装好所有函数即可得到，动态显示模拟遗传算法的效果图了，以下是完整代码：

from __future__ import division
import math
import numpy as np
import matplotlib.pyplot as plt
from matplotlib.animation import FuncAnimation
from matplotlib import animation



fig = plt.figure(figsize=(15,10))
ax1 = fig.add_subplot(111)
ax1.set_xlim([0,10])
ax1.set_ylim([-10,15])

#定义函数
def initFunction(x):
    y = x + 5*math.sin(5*x)+2*math.cos(3*x)
    return y

#形成一些点用于绘图（这里生成900个点）
x = [i / 100 for i in range(900)]
y = [i%900 for i in range(900)]
for i in range(900):
    y[i] = initFunction(x[i])
ax1.plot(x,y)


#初始化种群数(假设初始种群数目为10个)，10个17位位str
population = []
for i in range(20):
    strCode = ''
    for j in range(17):
        strCode += str(np.random.randint(0,2))
    population.append(strCode)

#解码
def decode(genCode):
    y = 0 + int(genCode,2)/(2**17-1)*9    
    return y

m = [m for m in range(len(population))]
n = [n for n in range(len(population))]
for i in range(len(population)):
    n[i] = initFunction(decode(population[i]))
    m[i] = decode(population[i])

sca1 = ax1.scatter(m,n)#散点图命名，用于更新

#
#适应度函数，其中负数置零的步骤可以调整
def fitnessfunction(population,initFunction):
    value = []
    for i in range(len(population)):
        value.append(initFunction(decode(population[i])))
        if value[i] < 0:
            value[i] = 0
    return value




#采用轮盘赌算法进行选择
#输入是种群的适应度函数值向量
#population 为父代，经过选择后返回子代childPopulation
def selectChild(population,fitnessResult):
    childPopulation = []
    Sumfitness = []
    indiviSumP = []
    for i in range(len(fitnessResult)):
        if i == 0:
            Sumfitness.append(fitnessResult[i])
        else:
            Sumfitness.append(Sumfitness[i-1] + fitnessResult[i])   
    for j in range(len(Sumfitness)):
        indiviSumP.append(Sumfitness[j]/sum(fitnessResult))
    for childNum in range(len(population)):
        x = np.random.uniform(0,1)
        if x <= indiviSumP[0]:
            childPopulation.append(population[0])
        else:
            for m in range(1,len(population)):
                if indiviSumP[m-1] <= x and x <= indiviSumP[m]:
                    childPopulation.append(population[m])
                    break     
    return childPopulation


#对选择后的子代进行一定概率的染色提交换
def crossParent(new_population,pc):
    crossChildPopulation = []
    #如果为单身则先选择出单shen
    if len(new_population)%2 != 0:
        randmSingleNum = np.random.randint(0,len(new_population))
        crossChildPopulation.append(new_population[randmSingleNum])
        del(new_population[randmSingleNum])

    half = int(len(new_population)/2)
    father = new_population[:half]
    mother = new_population[half:]
    np.random.shuffle(father)
    np.random.shuffle(mother)

    for i in range(half):
        crossP = np.random.uniform(0,1)
        if pc >= crossP:
            breakPoint = np.random.randint(0,len(new_population[0]))
            son = father[i][:breakPoint] + mother[i][breakPoint:]
            daughter = mother[i][:breakPoint] + father[i][breakPoint:]
        else:
            son = father[i]
            daughter = mother[i]
        crossChildPopulation.append(son)
        crossChildPopulation.append(daughter)
    return crossChildPopulation




#变异
def variation(crossChildPopulation,pc):
    variationPopulation = []
    for i in range(len(crossChildPopulation)):
        variationP = np.random.uniform(0,1)
        variationPos = np.random.randint(0,len(crossChildPopulation[0]))
        if variationP <= pc:
            if variationPos == 0:
                if crossChildPopulation[i][0] == '0':
                    crossChildPopulation[i] = '1'+crossChildPopulation[i][1:]
                else:
                    crossChildPopulation[i] = '0'+crossChildPopulation[i][1:]
            else:
                if crossChildPopulation[i][variationPos] == '0':
                    crossChildPopulation[i] = crossChildPopulation[i][:variationPos-1] + '1' + crossChildPopulation[i][variationPos:]
                else:
                     crossChildPopulation[i] = crossChildPopulation[i][:variationPos-1] + '0' + crossChildPopulation[i][variationPos:]
        variationPopulation.append(crossChildPopulation[i])
    return variationPopulation




def updata(i):
    global population
    x1 = [0 for j in range(len(population))]
    y1 = [0 for j in range(len(population))]
    fig.set_size_inches(15,10)#控制再保存图片的时候窗口不要缩

    fitnessValue = fitnessfunction(population,initFunction)
    selectChildes = selectChild(population,fitnessValue)
    crossChild = crossParent(selectChildes,0.7)
    population = variation(crossChild,0.02)
    for i in range(len(population)):
        x1[i] = decode(population[i])
        y1[i] = initFunction(decode(population[i]))
    data = [[x,y] for x,y in zip(x1,y1)]
    sca1.set_offsets(data)
    print(list(zip(x1,y1)))
    return sca1




ani = animation.FuncAnimation(fig = fig,func=updata,frames=np.arange(1,20),init_func = None,interval = 500,blit = False)
plt.show()

源代码链接：
链接：https://pan.baidu.com/s/1WL14a5-Q3tsqoIuJSikx4A 密码：ggu0